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Theorem dilsetN 37395
Description: The set of dilations for a fiducial atom 𝐷. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
dilset.a 𝐴 = (Atoms‘𝐾)
dilset.s 𝑆 = (PSubSp‘𝐾)
dilset.w 𝑊 = (WAtoms‘𝐾)
dilset.m 𝑀 = (PAut‘𝐾)
dilset.l 𝐿 = (Dil‘𝐾)
Assertion
Ref Expression
dilsetN ((𝐾𝐵𝐷𝐴) → (𝐿𝐷) = {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)})
Distinct variable groups:   𝑥,𝑓,𝐾   𝑓,𝑀   𝑥,𝑆   𝐷,𝑓,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑓)   𝐵(𝑥,𝑓)   𝑆(𝑓)   𝐿(𝑥,𝑓)   𝑀(𝑥)   𝑊(𝑥,𝑓)

Proof of Theorem dilsetN
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 dilset.a . . . 4 𝐴 = (Atoms‘𝐾)
2 dilset.s . . . 4 𝑆 = (PSubSp‘𝐾)
3 dilset.w . . . 4 𝑊 = (WAtoms‘𝐾)
4 dilset.m . . . 4 𝑀 = (PAut‘𝐾)
5 dilset.l . . . 4 𝐿 = (Dil‘𝐾)
61, 2, 3, 4, 5dilfsetN 37394 . . 3 (𝐾𝐵𝐿 = (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}))
76fveq1d 6664 . 2 (𝐾𝐵 → (𝐿𝐷) = ((𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)})‘𝐷))
8 fveq2 6662 . . . . . . 7 (𝑑 = 𝐷 → (𝑊𝑑) = (𝑊𝐷))
98sseq2d 3986 . . . . . 6 (𝑑 = 𝐷 → (𝑥 ⊆ (𝑊𝑑) ↔ 𝑥 ⊆ (𝑊𝐷)))
109imbi1d 345 . . . . 5 (𝑑 = 𝐷 → ((𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)))
1110ralbidv 3192 . . . 4 (𝑑 = 𝐷 → (∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥) ↔ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)))
1211rabbidv 3466 . . 3 (𝑑 = 𝐷 → {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)} = {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)})
13 eqid 2824 . . 3 (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}) = (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)})
144fvexi 6676 . . . 4 𝑀 ∈ V
1514rabex 5222 . . 3 {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)} ∈ V
1612, 13, 15fvmpt 6760 . 2 (𝐷𝐴 → ((𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)})‘𝐷) = {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)})
177, 16sylan9eq 2879 1 ((𝐾𝐵𝐷𝐴) → (𝐿𝐷) = {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  {crab 3137  wss 3920  cmpt 5133  cfv 6344  Atomscatm 36505  PSubSpcpsubsp 36738  WAtomscwpointsN 37228  PAutcpautN 37229  DilcdilN 37344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-dilN 37348
This theorem is referenced by:  isdilN  37396
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